Determining the output port from the distance

Rom. Rep. Phys. 72, 115 (2020) In this paper we discuss the following scenario. Suppose that two distant observers share an entangled state of two $D$-level systems. The two observers, Alice and Bob, have to send randomly their particles through a multi-input multi-output device. The task is to determine the nature of this device or black-box, such that Alice will be able to determine with certainty the output port of Bob's particle. We prove that this black-box is the multi-input-port quantum sorter. Further, we investigate if it is possible to determine the output port in the case when three observers are involved by analyzing the cases when the Greenberger-Horne-Zeilinger (GHZ) state and the W state are used.